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366 Capítulo 5 Polinomios y funciones polinomiales Práctica de habilidades Resuelva. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 4, 8 63. 64. 65. 66. 67. 68. 69. Para f(x) � 3x2 � 7x � 9, determine todos los valores de a para los que f(a) � 7. 70. Para f(x) � 4x2 � 11x � 2, determine todos los valores de a para los que f(a) � �4. 71. Para g(x) � 10x2 � 31x � 16, determine todos los valores de a para los que g(a) � 1. 72. Para g(x) � 6x2 � x � 3, determine todos los valores de a para los que g(a) � �2. 73. Para r(x) � x2 � x, determine todos los valores de a para los que r(a) � 30. 74. Para r(x) � 10x2 � 19x � 5, determine todos los valores de a para los que r(a) � �11. Utilice factorización para determinar las intersecciones con el eje x de las gráficas de cada ecuación (vea el ejemplo 10). 75. 76. 77. 78. 79. 80. Triángulo rectángulo En los ejercicios 81 a 86, utilice el teorema de Pitágoras para determinar x. 81. 82. 83. x � 8 x � 7 xx � 10 x � 3 x � 11 x � 2 x � 3x � 4 y = 12x3 - 39x2 + 30xy = 12x3 - 46x2 + 40x y = 15x2 - 14x - 8y = x2 + 16x + 64 y = x2 - 13x + 42y = x2 - 10x + 24 1a - 1213a + 22 = 4a 2b3 + 16b2 = -30b41a2 - 32 = 6a + 41a + 326a2 - 12 - 4a = 19a - 32 1x + 122 - 3x = 712x + 522 - 9 = 01x - 622 - 4 = 0 1x + 722 - 16 = 0-x2 + 16x = 63-x2 = 2x - 99 3x4 - 48x2 = 04y3 - 36y = 049c2 = 81 4x2 = 92y2 = 98x2 - 25 = 0 12a3 = 16a2 + 3a18z3 = 15z2 + 12z3b3 - 8b2 - 3b = 0 4c3 + 4c2 - 48c = 0x3 = -19x2 + 42xx3 - 3x2 = 18x 2a2 + 18a + 40 = 03x2 - 6x - 72 = 03a2 = -a + 2 2y2 = -y + 6a1a + 22 = 4812x + 521x - 12 = 12x c2 - 10c = -25x2 + 8x + 16 = 0b2 + b - 72 = 0 x2 + x - 12 = 0x2 - 6x + 5 = 0a2 + 6a + 5 = 0 18a2 = -36a3x2 = 27x-3x2 - 21x = 0 -x2 + 6x = 02a2 - 8a = 0x2 + 5x = 0 3y2 = -24y4x2 = 12x12x + 3214x + 52 = 0 13x - 2217x - 12 = 02a1a + 321a + 82 = 0x1x - 921x - 42 = 0 31a - 521a + 22 = 021x + 121x - 72 = 08x1x + 62 = 0 4x1x - 12 = 0x1x - 42 = 0x1x + 32 = 0 Sección 5.8 Ecuaciones polinomiales 367 84. 85. 86. Resolución de problemas En los ejercicios 87 a 90, determine las intersecciones con el eje x de cada gráfica; luego relacione la ecuación con la gráfica apropiada, marcada con las letras a) a d). 87. 88. 89. 90. y = x2 + x - 6y = x2 + 5x + 6y = x2 - x - 6y = x2 - 5x + 6 x � 9 x � 2 x � 11 x � 8 x � 1 x � 6 x � 8 x � 9 x � 1 a) b) c) d) Escriba una ecuación cuya gráfica tenga las intersecciones con el eje x en los valores dados. 91. 1 y 5. 92. 3 y �7. 93. 4 y �2. 94. y 6. 95. y 2. 96. y 2.6. 97. Mesa rectangular para café Una mesa para café es rectan- gular; si el largo mide 1 pie más que el doble de su ancho, y el área superficial de la tabla superior mide 10 pies cuadra- dos, determine su largo y ancho. 98. Cobertizo rectangular El piso de un cobertizo tiene un área de 60 pies cuadrados. Determine el largo y ancho, si el largo mide dos pies menos que el doble de su ancho. 99. Vela triangular La vela de un bote es triangular y su altura mide seis pies más que su base. Si el área de la vela es 80 pies cuadrados, determine su base y su altura. 100. Tienda triangular Una tienda de campaña triangular tiene una altura que mide 4 pies menos que su base. Si el área de un lado es 70 pies cuadrados, determine la base y la altura de la tienda. 101. Rectángulo El jardín de Frank Bullock está rodeado por un pasillo de ancho uniforme. El jardín y el pasillo juntos cu- bren un área de 320 pies cuadrados. Si el jardín mide 12 pies por 16 pies, determine el ancho del pasillo. 16 pies 12 pies -0.4 - 5 6 3 2 102. Marco de una pintura El marco de una pintura mide 28 cm por 23 cm. El área de la pintura es de 414 centímetros cua- drados. Determine el ancho del marco. 103. Hortaliza La hortaliza de Sally Yang es rectangular y mide 20 pies por 30 pies.Además para cubrir el terreno con man- tillo, ella quiere hacer un pasillo de ancho uniforme alrede- dor de la hortaliza. Si ella tiene suficiente mantillo para cubrir un área de 936 pies cuadrados, ¿cuál debe ser el an- cho del pasillo? 104. Jardín cuadrado Ronnie Tucker tiene un jardín cuadrado, a cuyo alrededor agrega un pasillo de 2 pies de ancho. Si el área total del pasillo y el jardín es de 196 pies cuadrados, determine las dimensiones del jardín. 105. Esculturas de agua En un edificio en Navy Pier en Chica- go, una fuente de agua, dispara pequeños chorros sobre un pasillo. Los chorros de agua alcanzan una altura máxima, y luego caen en un estanque al otro lado del pasillo. La altura respecto del disparador, h, de un chorro de agua t segundos después de que sale puede determinarse mediante la fun- 28 cm 23 cm x x 368 Capítulo 5 Polinomios y funciones polinomiales ción h(t) � �16t2 � 32t. Determine el tiempo que le toma al chorro de agua regresar a la altura del disparador; esto es, cuando h(t) � 0. 106. Proyectil Un modelo de cohete será lanzado desde una colina que se encuentra a 80 pies sobre el nivel del mar. El lugar del lanzamiento está cercano al océano (nivel del mar) y el cohete caerá en él. La distancia del cohete, s, por encima del nivel del mar en cualquier instante, t, se determi- na mediante la ecuación s(t) � �16t2 � 64t � 80. Determine el tiempo que le toma al cohete para caer en el océano. 107. Paseo en bicicleta Dos ciclistas, Bob y Tim, inician su paseo en el mismo punto. Bob conduce hacia el oeste y Tim hacia el norte. En algún momento, se encuentran separados entre sí por una distancia de 13 millas. Si Bob recorrió 7 millas más que Antonio, determine la distancia que recorrió cada uno de ellos. 108. Marco para pintura Abril está haciendo un marco para una pintura rectangular que le regalará a su mamá. La diagonal del marco es de 20 pulgadas. Determine las dimensiones del marco, si su longitud mide 4 pulgadas más que su ancho. 109. Cables de una tienda de campaña Una tienda de campaña se estabiliza mediante cables. Un cable se sujeta al suelo a 12 pies de distancia de la tienda. La longitud del cable utili- zado mide 8 pies más que la altura a donde se sujeta el otro extremo del cable. ¿Cuál es la longitud del cable? 12 x x � 8 Bob 13 milla s Tim 110. Automóvil atascado en el barro Suponga que dos automó- viles, indicados mediante los puntos A y B en la figura, jalan a un tercer automóvil, C, fuera del barro. Determine la dis- tancia del automóvil A al B. 111. Tienda de bicicletas Una tienda de bicicletas utiliza una ecua- ción para calcular sus ingresos mensuales, R(x) � 70x � x2, y otra para determinar sus costos mensuales, C(x) � 17x � 150, en donde x es el número de bicicletas vendidas y x � 10. De- termine el número de bicicletas que debe vender la tienda para alcanzar el punto de equilibrio (no ganar ni perder); esto es, el punto en donde los ingresos son iguales a los costos. 112. Flores de seda Edith Hall fabrica flores de seda y las vende a diferentes tiendas. Su compañía tiene una ecuación para calcular sus ingresos, R(x) � 40x � x2, y otra para determi- nar sus costos, C(x) � 14x � 25, en donde x es el número de flores vendidas y x � 5. Determine el número de flores que debe venderse para alcanzar el punto de equilibrio. 113. Fabricación de una caja Monique Sidding fabrica una caja cortando piezas de 2 por 2 pulgadas de un cartón cuadrado, doblando hacia arriba los lados, para crear una caja de 2 pulgadas de altura. ¿Cuál es el tamaño del cartón necesario para fabricar una caja de 2 pulgadas de altura con un volumen de 162 pulgadas cúbicas? 114. Fabricación de una caja Una caja rectangular se formará cortando cuadrados de cada esquina de una pieza rectangu- lar de hojalata y doblando hacia arriba los lados. La caja tendrá 3 pulgadas de altura, el largo será el doble del ancho, y el volumen será de 96 pulgadas cúbicas. Determine el lar- go y el ancho de la caja. 115. Cubo A un cubo sólido con dimensiones a3, se le quita un sólido rectangular condimensiones ab2. a) Escriba una fórmula para determinar el volumen que queda, V. b) Factorice el lado derecho de la fórmula de la parte a). c) Si el volumen es de 1620 pulgadas cúbicas y a es igual a 12 pulgadas, determine b. b b a a a 2 pulg. 2 pulg. C 15 piesA B x x 2x � 1 2x � 1 Sección 5.8 Ecuaciones polinomiales 369 116. Hoja de una sierra circular Una sierra circular de acero tie- ne un agujero en su centro, como se muestra en la figura. a) Escriba una fórmula para calcular el área de la hoja. b) Factorice el lado derecho de la fórmula de la parte a). c) Determine A, si R � 10 cm y r � 3 cm. 117. Considere la gráfica siguiente de una función cuadrática. a) Escriba una función cuadrática que tenga las intersec- ciones con el eje x indicadas. b) Escriba una ecuación cuadrática con una variable cuya solución sea �2 y �5. c) ¿Cuántas funciones cuadráticas diferentes pueden te- ner intersecciones con el eje x de �2 y �5? Explique. d) ¿Cuántas ecuaciones cuadráticas diferentes con una va- riable pueden tener soluciones de �2 y �5? Explique. �4 �2 �1 4 2 �3 3 1 1�7�6 �4�3 �1 y x R r 118. La gráfica de la ecuación y � x2 � 4 se ilustra a continuación. a) ¿Cuántas intersecciones con el eje x tiene la gráfica? b) ¿Cuántas soluciones reales tiene la ecuación x2 � 4 � 0? Explique su respuesta. 119. Considere la función cuadrática a) La gráfica de este tipo de función puede no tener inter- secciones con el eje x, una intersección con el eje x o dos intersecciones con el eje x. Bosqueje cada una de estas posibilidades.ee graphing answer sección, b) ¿Cuántas posibles soluciones reales puede tener la ecuación ax2 � bx � c � 0, a � 0? Explique su respues- ta a la parte b) utilizando los bosquejos de la parte a). 120. Distancia para detenerse La distancia, d en pies, para dete- ner un automóvil común que viaja sobre pavimento seco puede calcularse mediante la función d(s) � 0.034s2 � 0.56s � 17.11, en donde s es la velocidad del automóvil antes de frenar y 60 � s � 80 millas por hora. Si un automóvil re- quiere de 190 pies para detenerse después de aplicar los frenos, ¿qué tan rápido va el automóvil? 121. Distancia para detenerse La distancia, d en pies, para dete- ner un automóvil común que viaja sobre pavimento mojado puede calcularse mediante la función d(s) � �0.31s2 � 59.82s � 2180.22, en donde s es la velocidad del automóvil antes de frenar y 60 � s � 80 millas por hora. Si un automó- vil requiere de 545 pies para detenerse después de aplicar los frenos, ¿qué tan rápido va el automóvil? P1x2 = ax2 + bx + c, a 7 0. x y �2 2 8 6 4 2 4�2�4 Retos Resuelva. 122. 123. 124. 2, 1 Actividad en grupo En cursos más avanzados de matemáticas podría necesitar despejar (se lee “y prima”) en una ecuación. Cuando esto ocurra, trate a como una variable diferente de y. De forma individual despeje a de cada ecuación. En equipo, compare sus respuestas y obtenga las respuestas correctas en equipo. 125. 126. 127. Ejercicios de repaso acumulativo 2xyy¿ - xy = x - 3y¿xy - xy¿ = 3y¿ + 2xy¿ + yy¿ = 1 y¿y¿ y¿ x6 - 9x3 + 8 = 0x4 - 13x2 = -36x4 - 17x2 + 16 = 0 [1.5] 128. Simplifique (4x�2y3)�2. [2.5] 129. Resuelva la desigualdad y grafique la solución en la recta numérica. -1 6 413x - 22 3 … 5 [4.1] 130. Resuelva el sistema de ecuaciones [5.2] 131. Si f(x) � �x2 � 3x y g(x) � x2 � 5, determine [5.7] 132. 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<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>>> >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Gray Gamma 1.8) /CalRGBProfile (Apple RGB) /CalCMYKProfile (U.S. Sheetfed Uncoated v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.4 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJDFFile false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false 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